Regular megagon  

A regular megagon


Type  Regular polygon 
Edges and vertices  1000000 
Schläfli symbol  {1000000}, t{500000}, tt{250000}, ttt{125000}, tttt{62500}, ttttt{31250}, tttttt{15625} 
Coxeter diagram  
Symmetry group  Dihedral (D_{1000000}), order 2×1000000 
Internal angle (degrees)  179.99964° 
Dual polygon  self 
Properties  convex, cyclic, equilateral, isogonal, isotoxal 
A megagon is a polygon with 1 million sides (mega, from the Greek μέγας megas, meaning "great").^{[1]}^{[2]} Even if drawn at the size of the Earth, a regular megagon would be very difficult to distinguish from a circle.
Regular megagon
A regular megagon is represented by Schläfli symbol {1000000} and can be constructed as a truncated 500000gon, t{500000}, a twicetruncated 250000gon, tt{250000}, a thricetruncated 125000gon, ttt{125000), or a fourfoldtruncated 62500gon, tttt{62500}, a fivefoldtruncated 31250gon, ttttt{31250}, or a sixfoldtruncated 15625gon, tttttt{15625}.
A regular megagon has an interior angle of 179.99964°.^{[1]} The area of a regular megagon with sides of length a is given by
The perimeter of a regular megagon inscribed in the unit circle is:
which is very close to 2π. In fact, for a circle the size of the Earth's equator, with a circumference of 40,075 kilometres, one edge of a megagon inscribed in such a circle would be about 40 meters long. The difference between the perimeter of the inscribed megagon and the circumference of this circle comes to less than 1/16 millimeters.^{[3]}
Because 1000000 = 2^{6} × 5^{6}, the number of sides is not a product of distinct Fermat primes and a power of two. Thus the regular megagon is not a constructible polygon. Indeed, it is not even constructible with the use of neusis or an angle trisector, as the number of sides is neither a product of distinct Pierpont primes, nor a product of powers of two and three.
Philosophical application
Like René Descartes' example of the chiliagon, the millionsided polygon has been used as an illustration of a welldefined concept that cannot be visualised.^{[4]}^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}^{[10]}
The megagon is also used as an illustration of the convergence of regular polygons to a circle.^{[11]}
Symmetry
The regular megagon has Dih_{1000000} dihedral symmetry, order 2000000, represented by 1000000 lines of reflection. Dih_{100} has 48 dihedral subgroups: (Dih_{500000}, Dih_{250000}, Dih_{125000}, Dih_{62500}, Dih_{31250}, Dih_{15625}), (Dih_{200000}, Dih_{100000}, Dih_{50000}, Dih_{25000}, Dih_{12500}, Dih_{6250}, Dih_{3125}), (Dih_{40000}, Dih_{20000}, Dih_{10000}, Dih_{5000}, Dih_{2500}, Dih_{1250}, Dih_{625}), (Dih_{8000}, Dih_{4000}, Dih_{2000}, Dih_{1000}, Dih_{500}, Dih_{250}, Dih_{125}, Dih_{1600}, Dih_{800}, Dih_{400}, Dih_{200}, Dih_{100}, Dih_{50}, Dih_{25}), (Dih_{320}, Dih_{160}, Dih_{80}, Dih_{40}, Dih_{20}, Dih_{10}, Dih_{5}), and (Dih_{64}, Dih_{32}, Dih_{16}, Dih_{8}, Dih_{4}, Dih_{2}, Dih_{1}). It also has 49 more cyclic symmetries as subgroups: (Z_{1000000}, Z_{500000}, Z_{250000}, Z_{125000}, Z_{62500}, Z_{31250}, Z_{15625}), (Z_{200000}, Z_{100000}, Z_{50000}, Z_{25000}, Z_{12500}, Z_{6250}, Z_{3125}), (Z_{40000}, Z_{20000}, Z_{10000}, Z_{5000}, Z_{2500}, Z_{1250}, Z_{625}), (Z_{8000}, Z_{4000}, Z_{2000}, Z_{1000}, Z_{500}, Z_{250}, Z_{125}), (Z_{1600}, Z_{800}, Z_{400}, Z_{200}, Z_{100}, Z_{50}, Z_{25}), (Z_{320}, Z_{160}, Z_{80}, Z_{40}, Z_{20}, Z_{10}, Z_{5}), and (Z_{64}, Z_{32}, Z_{16}, Z_{8}, Z_{4}, Z_{2}, Z_{1}), with Z_{n} representing π/n radian rotational symmetry.
John Conway labels these lower symmetries with a letter and order of the symmetry follows the letter.^{[12]} r2000000 represents full symmetry and a1 labels no symmetry. He gives d (diagonal) with mirror lines through vertices, p with mirror lines through edges (perpendicular), i with mirror lines through both vertices and edges, and g for rotational symmetry.
These lower symmetries allows degrees of freedom in defining irregular megagons. Only the g1000000 subgroup has no degrees of freedom but can seen as directed edges.
Megagram
A megagram is a millionsided star polygon. There are 199,999 regular forms^{[13]} given by Schläfli symbols of the form {1000000/n}, where n is an integer between 2 and 500,000 that is coprime to 1,000,000. There are also 300,000 regular star figures in the remaining cases.
References
 ^ ^{a} ^{b} Darling, David J., The universal book of mathematics: from Abracadabra to Zeno's paradoxes, John Wiley & Sons, 2004. Page 249. ISBN 0471270474.
 ^ Dugopolski, Mark, College Algebra and Trigonometry, 2nd ed, AddisonWesley, 1999. Page 505. ISBN 0201347121.
 ^ Williamson, Benjamin, An Elementary Treatise on the Differential Calculus, Longmans, Green, and Co., 1899. Page 45.
 ^ McCormick, John Francis, Scholastic Metaphysics, Loyola University Press, 1928, p. 18.
 ^ Merrill, John Calhoun and Odell, S. Jack, Philosophy and Journalism, Longman, 1983, p. 47, ISBN 0582281571.
 ^ Hospers, John, An Introduction to Philosophical Analysis, 4th ed, Routledge, 1997, p. 56, ISBN 0415157927.
 ^ Mandik, Pete, Key Terms in Philosophy of Mind, Continuum International Publishing Group, 2010, p. 26, ISBN 1847063497.
 ^ Kenny, Anthony, The Rise of Modern Philosophy, Oxford University Press, 2006, p. 124, ISBN 0198752776.
 ^ Balmes, James, Fundamental Philosophy, Vol II, Sadlier and Co., Boston, 1856, p. 27.
 ^ Potter, Vincent G., On Understanding Understanding: A Philosophy of Knowledge, 2nd ed, Fordham University Press, 1993, p. 86, ISBN 0823214869.
 ^ Russell, Bertrand, History of Western Philosophy, reprint edition, Routledge, 2004, p. 202, ISBN 0415325056.
 ^ The Symmetries of Things, Chapter 20
 ^ 199,999 = 500,000 cases  1 (convex)  100,000 (multiples of 5)  250,000 (multiples of 2) + 50,000 (multiples of 2 and 5)
